L(s) = 1 | + 1.79i·2-s + (−0.240 − 1.71i)3-s − 1.22·4-s + (3.08 − 0.431i)6-s − 5.23·7-s + 1.39i·8-s + (−2.88 + 0.824i)9-s + 3.31i·11-s + (0.294 + 2.10i)12-s + 3.60·13-s − 9.39i·14-s − 4.94·16-s + (−1.48 − 5.17i)18-s − 7.00·19-s + (1.25 + 8.97i)21-s − 5.95·22-s + ⋯ |
L(s) = 1 | + 1.26i·2-s + (−0.138 − 0.990i)3-s − 0.612·4-s + (1.25 − 0.176i)6-s − 1.97·7-s + 0.492i·8-s + (−0.961 + 0.274i)9-s + 1.00i·11-s + (0.0849 + 0.606i)12-s + 1.00·13-s − 2.50i·14-s − 1.23·16-s + (−0.349 − 1.22i)18-s − 1.60·19-s + (0.274 + 1.95i)21-s − 1.26·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0324106 - 0.464615i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0324106 - 0.464615i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.240 + 1.71i)T \) |
| 11 | \( 1 - 3.31iT \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 - 1.79iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 5.23T + 7T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 - 6.19iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 7.55iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.65iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 17.5iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87220434245156212890859279002, −10.70636671702002045099883386433, −9.523849311081488977472079998799, −8.665653243040392346104393825594, −7.63032242191580243520027241901, −6.80843757629515273292576083519, −6.30965862367012349154723168965, −5.55670109304934954819462031661, −3.78671255189883856605112902232, −2.23272531935447420161228503637,
0.27692121190235620558100467852, 2.67950142751477985719572847679, 3.50633770236643508773230653860, 4.21015840115354118315708847823, 6.08817641109469990781126677222, 6.44641851954641807363361700480, 8.532357644977074963617994121070, 9.220443836702836402785948865080, 10.09621800988871643857728715912, 10.61340225112606574396655280605